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Volevič systems of singular nonlinear partial differential equations. (English) Zbl 0854.35123
The following system of singular nonlinear partial differential equations is studied: \[ \varphi(t) {\partial u_i\over \partial t}= f_i(t, x, (\mu_0(t) D^k u_j)_{(j, k)\in {\mathfrak N}(i)})+ g_i(t, x),\quad 1\leq i\leq N, \] where \(u= (u_1,\dots, u_N)\in Y^N\) and \(x\in \Omega\). Here \(\Omega\) is an open subset of \(X\); \(X\) is a Banach space; moreover, \(D\) stands for the Fréchet differentiation with respect to \(x\). The main result of the paper consists in the formulation and proving an existence and uniqueness theorem of the local solution in the ultradifferentiable class. The results obtained are interesting; however, the paper is very technical and includes no application.

35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
35A10 Cauchy-Kovalevskaya theorems
Full Text: DOI
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